Once addition has been explored a bit, it leads pretty naturally to a new question: if there are three bricks in a pile, how many bricks do I need to add to it so that there will be five in the pile?
Our addition problems were all phrased using a pattern like
number + number = what?
and the question above rearranges it a bit:
number + what? = number
what? + number = number
This question is usually asked as “What is the difference between what we have and our goal?”, which is much more compact and cleaner sounding than my introductory question above.
This way of asking the question introduces a more sophisticated idea than addition. Instead of looking at two piles of physical objects and contemplating joining them, we are looking at two piles and trying to use a number to describe what makes them different, their “difference”. Subtraction is the arithmetic operation used to find a quantitative difference.
Instead of writing the above problem as a sum with a missing piece,
we can write it as a subtraction problem
which is read as either “what is the difference between 10 and 3?” or “10 take away 3”. The answers to the addition and subtraction versions of the problem above will be the same, since both are descriptions of the same problem.
This leads to some interesting insights:
- Algebra provides more than one way of describing this problem
- Using English to describe problems with the unknown quantity in the middle (as occurs in the first three above) can be awkward. A phrase seems easier to follow, when said aloud, when the unknown quantity is all by itself on one side of the equal sign.
Properties of subtraction
Subtraction provides a slightly more convenient way of describing this problem than addition does. Let’s investigate how subtraction works a bit, to see if the insights we gained about addition will work here too.
Is subtraction associative? If it is, then the following quantities should all be equal:
The first, evaluating it from left to right , produces a result of 1. The second also produces a result of 1, and note that it had us performing the computations in exactly the same order as the first one. The third one produces a different result of 5. Note that it is the only one that had us compute things in a different order… so order matters with subtraction, and the “Order of Operations” requirement that “in the event of a tie we compute from left to right” is most necessary here.
Since we did not end up with the same result in all cases above, as we played around with associating different quantities in a subtraction problem, subtraction is not associative.
Is subtraction commutative? If it is, then the following quantities should all be equal:
Evaluating them all from left to right, the results are: 1, 1, -7, -7, -5, -5. While some results are the same, others are not. So, we cannot move the numbers around in a subtraction problem without risk of changing the answer. Subtraction is not commutative.
Compared to addition, subtraction is inflexible. The rule about evaluating things “from left to right” in the order of operations must be followed to arrive at the intended answer when working on subtraction problems.
Furthermore, something interesting happened above as we investigated whether subtraction is commutative or not. Even though our problem started out with nice simple counting numbers like 6, 3, and 2 (just as our earlier addition problems did) the answers to these subtraction problems have forced us to invent new numbers that we never needed when playing with addition: 0, -1, -2, -3, -4, -5, -6, -7, etc.
These numbers do not correspond to anything we saw or thought of while joining piles of bricks (adding). What do they mean, and can we make use of them?
And finally, the word “difference” does not mean exactly the same thing as “subtract”. If you ask ten people on a street corner to find the difference between 10 and 3, or between 3 and 10, they will probably all answer “seven” – no matter which way the question was phrased. Yet when asked to subtract 10 from 3 they will all answer “negative seven”, and when asked to subtract 3 from 10 they will all answer “seven”.
When asked to find a difference we intuitively expect a positive number, the amount by which two quantities differ, as the answer. So, we feel free (outside of math class) to arrange and solve the problem in such a way that it will produce a positive result. However, when asked to subtract one number from another, we have no flexibility in how we set up the problem – there is only one way to carry out the task. Choose your words carefully!
Yet, the phrasing of problems in a math text will often refer to something like “the difference between two and five”, and expect us to treat it like we would subtraction: start with a value of two then take away five, ending up with a negative three.
Earlier posts in this series:
Algebra Intro 1: Numbers and Variables
Algebra Intro 2: Addition
Posts that continue this series:
Algebra Intro 3: Subtraction
Algebra Intro 4: Negative Numbers, Zero, Absolute Values, and Opposites
Algebra Intro 5: Addition, Subtraction, and Terms
Algebra Intro 6: Multiplication
Algebra Intro 7: Properties of Multiplication
Algebra Intro 8: Division
Algebra Intro 9: Fractions, Reciprocals, and Properties of Division
Algebra Intro 10: Fractions and Multiplication
Algebra Intro 11: Dividing Fractions, Equivalent Fractions
Algebra Intro 12: Adding and Subtracting Fractions